Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 16

(

A ( k 1 , l 1 ) ) ( k 2 , l 2 ) = ( A ( k 2 , l 2 ) ) ( k 1 , l 1 ) .

Third, let P ={ k 1 , k 2 ,..., k s }⊆ K and Q ={ q 1 , q 2 ,..., q t }⊆ L. Then, we

define the following three operations:

A

) = (...((

A

) ) ( k 2 , l ) )...) ( k s , l ) , ( P , l ( k 1 , l

A

) = (...((

A

l 1 ) ) ( k , l 2 ) )...) ( k , l t ) ,

( k , Q ( k ,

A

) = (...((

A

) ) ( p 2 , Q ) )...) ( p s , Q ) = (. . . ((

A

) ) ( P , q 2 ) )...) ( P , q t ) .

( P , Q ( p 1 , Q ( P , q 1

Obviously,

A

) =

I and ]] A

( ∅ , ∅ ) =

A .

( K , L ∅

Theorem 3 For every two IMs A =[ K , L , { a k i , l j }] , B =[ P , Q , { b p r , q s }] :

A ⊆ d B fA = B ( P − K , Q − L ) .

Proof Let A ⊆ d B . Therefore, K ⊆ P and L ⊆ Q and for every k ∈ K , l ∈ L :

a k , l = b k , l . From the definition,

B ( P − K , Q − L ) = (...(( B ( p 1 , q 1 ) ) ( p 1 , q 2 ) )...) ( p r , q s ) ,

where p 1 , p 2 ,..., p r ∈ P − K , i.e., p 1 , p 2 ,..., p r ∈ P , and p 1 , p 2 ,..., p r ∈ K ,

and q 1 , q 2 ,..., q s ∈ Q − L , i.e., q 1 , q 2 ,..., q s ∈ Q , and q 1 , q 2 ,..., q s ∈ L .

Therefore,

B

) =[ P − ( P − K ), Q − ( Q − L ), { b k , l }] = [ K , L , { b k , l }] = [ K , L , { a k , l }] = A ,

( P − K , Q − L

because by definition the elements of the two IMs, which are indexed by equal
symbols, coincide.

For the opposite direction we obtain, that if A = B , then

( P − K , Q − L )

A = B

) ⊆ d B ∅ , ∅ =

B .

( P − K , Q − L

1.7 Operation “Projection” Over an R -IM, ( 0 , 1 ) -IM and L-IM

Let M ⊆ K and N ⊆ L . Then,

pr M , N A =[ M , N , { b k i , l j }] ,